Point at Infinity

is the point on the line such that . It can also be thought of as the point of intersection of two parallel lines. In 1639, Desargues (1864) became the first to consider the point at infinity (Cremona 1960, p. ix), although Poncelet was the first to systematically employ the point at infinity (Graustein 1930).

A point lying on the line at infinity is a point at infinity. In particular, a point with trilinear coordinates is a point at infinity if it satisfies

Points at infinity therefore do not have exact trilinear coordinates.

In projective geometry, the choice of a line at infinity is the choice of an affine chart. Consequently, a projective collineation can move a point that is at infinity in one chart to a finite point in another. A statement that a finite configuration contains a point at infinity therefore generally describes a particular affine drawing, rather than an invariant property of the projective configuration.

Terminology for geometric realizations of point-line configurations is not uniform. Bokowski and Pilaud (2014) call a realization by straight lines in the real projective plane geometric, allowing points at infinity. Berman et al. (2024) use strong geometric realization for distinct points and straight lines in Euclidean space, usually the Euclidean plane, which excludes points at infinity.

Kimberling centers are points at infinity for (the Euler infinity point), 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 674, 680, 681, 688, 690, 696, 698, 700, 702, 704, 706, 708, 710, 712, 714, 716, 718, 720, 722, 724, 726, 730, 732, 734, 736, 740, 742, 744, 746, 752, 754, 758, 760, 766, 768, 772, 776, 778, 780, 782, 784, 786, 788, 790, 792, 794, 796, 802, 804, 806, 808, 812, 814, 816, 818, 824, 826, 830, 832, 834, 838, 888, 891, 900, 912, 916, 918, 924, 926, 928, 952, 971, 1154, 1499, 1503, 1510, 1511, 1912, 1938, 1946, 2385, 2386, 2387, 2388, 2389, 2390, 2391, 2392, and 2393 (Weisstein, Oct. 25 and Nov. 20, 2004).

The term point at infinity is also used for complex infinity (Krantz 1999, p. 82).